Ramanujan Graphs and Interlacing Families

TL;DR

This survey addresses the problem of constructing Ramanujan graphs for broad degrees and understanding spectra of random covers. It centers on the interlacing families method, relating the expected characteristic polynomials to eigenvalues and connecting to matching polynomials and free probability via Walsh convolution. Key contributions include: (i) establishing bipartite Ramanujan graphs for all degrees through 2-covers, (ii) extending to $n$-covers and representation signings, (iii) handling beyond-tree universal covers via additive product graphs, and (iv) deriving nontrivial probabilistic Ramanujan bounds for random regular graphs using finite free probability machinery; and (v) highlighting algorithmic aspects in specific models. The work broadens the toolkit at the intersection of spectral graph theory, combinatorics, and free probability, with implications for pseudorandomness and expander constructions.

Abstract

This survey accompanies a lecture on the paper ``Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees'' by A. Marcus, D. Spielman, and N. Srivastava at the 2024 International Congress of Basic Science (ICBS) in July, 2024. Its purpose is to explain the developments surrounding this work over the past ten or so years, with an emphasis on connections to other areas of mathematics. Earlier surveys about the interlacing families method by the same authors focused on applications in functional analysis, whereas the focus here is on applications in spectral graph theory.

Theorems & Definitions (14)

• Definition 1.1
• Theorem 1.2: Alon-Boppana alon1987monotone
• Remark 1.3
• Theorem 1.4: lubotzky1988ramanujanmargulis1988explicit
• Theorem 2.1: Interlacing Families
• Lemma 2.2
• Theorem 3.1
• Theorem 3.2: Godsil-Gutman
• Theorem 3.3: Heilmann-Lieb
• Theorem 3.4: hall2018ramanujan